In the previous section, we discussed how the cell reads its shape and imposes patterns by using MT-related force balances. Another group of mechanisms enabling cells to discriminate among all potential directions and to select a particular direction as being ‘special’ is based on reactions and diffusion of signaling molecules. This process, known as polarization, is fundamental to nearly all cells [

15]. Theoreticians view this as an example of ‘symmetry-breaking’, a class of phenomena where a chemically reactive system spontaneously generates spatially heterogeneous patterns. Pioneering work by Alan Turing in the 1950s showed that simple sets of reactions coupled to finite rates of diffusion could lead to spontaneous symmetry breaking [

16]. Subsequent theoretical studies in the 1960s and 1970s hypothesized that such reaction-diffusion mechanisms might underlie the morphogenesis of the early embryo [

17,

18]. Fueled by recent advances in fluorescent protein imaging (i.e. confocal and 2-photon imaging, better digital cameras, and better fluorophores) and in digital technology (i.e. faster processors, larger memory capacity), we are witnessing a rapid growth in the integrated experimental-theoretical study of polarization. Here, we discuss how modeling in 3-D is shaping advances in our understanding of cell polarity by examining the specific case of embryo polarization.

The polarization of early stage embryos is vital to determining the major axes of the mature organism: anterior-posterior, left-right, dorsal-ventral. Model organisms, such as

*C. elegans* and

*Drosophila melanogaster*, have been crucial to the study of embryo polarization, and decades of research in this field has facilitated mathematical modeling by providing a wealth of quantitative data within a consistent spatial-temporal framework (3-D spatially plus time reference frame, referred to hereafter as “3-D+T”). One of the best-studied polarity determinants is the protein bicoid, a transcriptional regulator that exists in an anterior-to-posterior concentration gradient in the early stages of

*D. melanogaster* embryonic development [

19]. Early work documented the existence of the bicoid gradient, and led to the synthesis-diffusion-degradation (SDD) model [

19]. In this model, bicoid protein is synthesized from maternally inherited bicoid mRNA located near the anterior end of the embryo, diffuses, and is randomly degraded (). Although the SDD was formulated without mathematics, it qualitatively described basic features of the bicoid gradient. Until recently, however, quantitative tests of the SDD model were lacking, so it was not clear whether this model was simply an attractive idea or a rigorous mechanistic explanation. For example, the SDD model predicts that at steady-state the gradient will decay according to an exponential distribution. Whether the bicoid gradient conforms to an exponential, or whether it even reaches a steady-state at all during developmentally relevant time scales were key questions left unanswered until recent years.

With the rapid advancement of digital cameras and fluorescent protein technology, the last few years have witnessed a flurry of new studies on bicoid, as well as other morphogens [

20]. In the case of bicoid, detailed quantitative analysis confirmed an approximately exponential decay in the concentration of bicoid, as measured in 1-D along the contour of the embryo’s periphery, consistent with a simple diffusion-degradation mechanism [

21]. For this simple model to work quantitatively, bicoid must diffuse at a rate sufficiently fast to allow the gradient to develop from a characteristic decay length of ~60 µm at 50 min after egg deposition to ~110 µm at 160 min [

22]. Using photobleaching of bicoid-GFP via two-photon confocal microscopy, the first direct estimate of the bicoid diffusion coefficient was deemed too small to yield the observed extent of the gradient according to the SDD model [

23]. This finding led to new modeling efforts to explain the additional missing convective transport mechanisms, perhaps via cytoskeleton-based transport [

24]. One could regard this as an “SDD+” model, where the “+” refers to some extra transport component that is as yet undiscovered. Interpretation of photobleaching experiments is notoriously tricky [

25,

26], however, and subsequent analysis using a full 3-D simulation of the bleaching-diffusion experiment re-estimated the measured value of the bicoid diffusion coefficient to be 3-fold higher than initially reported, and consistent with the SDD model after all [

22]. It is important to note, however, that this measurement was made at a specific time and place in the developmental process: the anterior cytoplasm during mitotic cycle 13. To really test models, it will be vital to measure the diffusion coefficient in 3-D space and time (3D+T), both during mitosis and interphase (and inside and outside of the nucleus). It may turn out that the diffusion coefficient, rather than being a constant parameter, is actually a variable that depends on space and time, as recently suggested [

27].

In fact, due to the combination of 3-D modeling and improved 3-D imaging, nearly all the basic assumptions of the SDD model have recently been thrown into question. For example, the first ‘D’ in the SDD model, ‘diffusion’, recently came into question when it was asserted, in the absence of mathematical modeling, that the bicoid protein gradient could be explained entirely by a pre-existing, maternally derived bicoid mRNA gradient [

28]. In this case there would not be a requirement for bicoid protein diffusion at all, other than from the mRNA-ribosomal synthesis site to its target binding sites in the nucleus. However, it seems that this model would have difficulty explaining the increase in the decay length of the gradient over time, mentioned above. In addition, a recent report shows that protein diffusion is necessary to explain the different decay lengths in the mRNA and protein, as measured quantitatively [

27]. An important aspect of this recent study is that it included a full 3-D model, using the experimentally measured embryo geometry as the domain over which the solution was obtained [

29]. Combined with comprehensive 3-D quantitative analysis, they were able to show that the bicoid protein gradient is not simply a rescaled version of the bicoid mRNA gradient. The last ‘D’ in the SDD model, degradation, is also controversial because it is not clear that degradation is actually occurring during the period when bicoid influences polarity development. In fact, recent modeling analysis makes the case for bicoid not being degraded at all [

30]. In this ‘nuclear-trapping’ model, net consumption of synthesized bicoid protein is provided by newly produced nuclei; as the number of nuclei increases the capacity of bicoid per unit volume of embryo also increases. Thus, the SDD model could in one respect be simpler than previously imagined in that no degradation is necessarily required, such that an ‘SD’ model (i.e. synthesis-diffusion) might be adequate.

In summary, it appears that understanding the dynamics of bicoid and other morphogens will require model predictions and measurements in 3-D+T to discriminate among various hypotheses and test long-held assumptions. For example, it may be important to incorporate even more realistic models of the cell geometry, such as the inward invaginations of the plasma membrane that would presumably limit diffusion on the 10–20 µm length scale [

31]. However, as a full 3-D+T picture emerges, it may still be possible to mathematically homogenize the cytoplasm and thereby return to a simpler 1-D+T model [

32,

33]. Such coarse-graining/homogenization methods are standard in engineering, and are commonly used to predict and simulate the electrical and diffusive transport properties of both naturally occurring and fabricated composite materials [

34–

36]. Reducing complexity by coarse-graining (i.e. from 3-D down to 2-D or 1-D) will make it easier to rapidly obtain

*in silico* predictions of mutant phenotypes, and thereby gain a more complete molecular-level mechanistic understanding of morphogen systems.